Pulse the cremini mushrooms in a food processor until they are very finely minced.
Sauté the minced mushrooms in a pan with olive oil, fresh thyme, sea salt, and black pepper over medium heat until all moisture has evaporated and the mixture is dry.
Season the beef tenderloin with sea salt and black pepper, then sear in a hot pan for approximately one minute per side until a brown crust forms.
Brush the seared beef tenderloin on all sides with the Dijon mustard while the meat is still warm to allow the flavors to penetrate.
Roll the puff pastry out into a thin rectangle on a lightly floured surface or piece of parchment paper.
Spread the mushroom duxelles evenly over the pastry, place the beef in the center, and wrap the pastry tightly around the meat, trimming any significant excess.
Place the Wellington on a baking sheet, brush the exterior with the egg white, and bake at 400°F for 15 to 20 minutes until the pastry is crisp and golden.